Ideal MHD. Part II: Rigidity from infinity for ideal Alfvén waves in 3D thin domains

Abstract

This paper concerns the rigidity from infinity for Alfvén waves governed by ideal incompressible magnetohydrodynamic equations subjected to strong background magnetic fields along the $x_1$-axis in 3D thin domains ${\Omega }_{\delta }={\mathbb{R}}^2\times (-\delta ,\delta )$ with $\delta \in (0,1]$ and slip boundary conditions. We show that in any thin domain ${\Omega }_{\delta }$, Alfvén waves must vanish identically if their scattering fields vanish at infinities. As an application, the rigidity of Alfvén waves in ${\Omega }_{\delta }$, propagating along the horizontal direction, can be approximated by the rigidity of Alfvén waves in ${\mathbb{R}}^2$ when $\delta$ is sufficiently small. Our proof relies on the uniform (with respect to $\delta$) weighted energy estimates with a position parameter in weights to track the center of Alfvén waves. The key issues in the analysis include dealing with the nonlinear nature of Alfvén waves and the geometry of thin domains.

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